(c) A Solid Weights 64N In Air And 48N When Totally Immersed In A Liquid 0.8g/cm3. Calculate The Upthrust
Understanding the concept of upthrust, also known as buoyant force, is fundamental in physics, particularly in the study of fluids and objects submerged within them. In this article, we will thoroughly analyze the problem involving a solid object, which weighs 64N in air, and weighs 48N when fully immersed in a liquid with a density of 0.8 g/cm³. Our goal is to calculate the upthrust acting on the object. We will explore the principles behind the calculation, provide step-by-step solutions, and discuss related concepts to deepen your understanding.
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Understanding the Problem Statement
Before diving into calculations, it is essential to comprehend the key components of the problem:
- Weight in Air (W_air): 64N
- Weight in Liquid (W_liquid): 48N
- Density of Liquid (ρ_liquid): 0.8 g/cm³
The question asks us to find the upthrust (also called buoyant force) acting on the solid object when fully immersed in the liquid.
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Key Concepts and Principles
1. Weight and Apparent Weight
- Actual weight (W): The force due to gravity acting on the object; it is constant irrespective of the medium.
- Apparent weight in a fluid: The weight measured when the object is immersed in a fluid; it is less than the actual weight due to the upward buoyant force.
2. Upthrust (Buoyant Force)
- The upthrust is the upward force exerted by the fluid on the submerged object.
- According to Archimedes’ principle:
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Calculating the Upthrust
Given the data, the process involves several steps:
Step 1: Determine the actual weight of the object
- The actual weight of the object in air is directly given:
Step 2: Find the apparent weight when immersed in the liquid
- The apparent weight when fully immersed:
Step 3: Calculate the upthrust using the difference in weights
- From the principle of apparent weight:
Substituting the known values:
\[ \text{Upthrust} = 64\, \text{N} - 48\, \text{N} = 16\, \text{N} \]
Therefore, the upthrust acting on the object is 16 N.
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Verifying the Calculation with Archimedes’ Principle
To ensure our calculation is accurate, let's verify using Archimedes’ principle, which states:
\[ \text{Upthrust} = \text{Weight of displaced fluid} \]
Given the density of the liquid and the volume of the object, we can cross-check our work.
Step 4: Find the volume of the object
Using the actual weight:
\[ W_{air} = mg \]
where:
- \( m \) is the mass of the object,
- \( g \) is acceleration due to gravity (\( \approx 9.8\, \text{m/s}^2 \)).
Thus:
\[ m = \frac{W_{air}}{g} = \frac{64\, \text{N}}{9.8\, \text{m/s}^2} \approx 6.53\, \text{kg} \]
Now, the volume (\( V \)) of the object:
\[ V = \frac{m}{\rho_{solid}} \]
But since the density of the solid is not directly provided, we can instead relate the upthrust to fluid parameters.
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Calculating the Volume of Displaced Fluid
Using the relation:
\[ \text{Upthrust} = \rho{liquid} \times V{displaced} \times g \]
Rearranged to find \( V_{displaced} \):
\[ V{displaced} = \frac{\text{Upthrust}}{\rho{liquid} \times g} \]
Convert the density of the liquid to SI units:
\[ \rho_{liquid} = 0.8\, \text{g/cm}^3 = 0.8 \times 10^3\, \text{kg/m}^3 = 800\, \text{kg/m}^3 \]
Now, substitute the values:
\[ V_{displaced} = \frac{16\, \text{N}}{800\, \text{kg/m}^3 \times 9.8\, \text{m/s}^2} \]
Calculating:
\[ V_{displaced} = \frac{16}{7840} \approx 0.00204\, \text{m}^3 \]
This volume corresponds to the volume of the object submerged, confirming that the upthrust equals the weight of the displaced fluid.
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Related Concepts and Considerations
1. Factors Affecting Upthrust
- Density of the fluid: Higher density results in greater buoyant force.
- Volume of the object submerged: The larger the submerged volume, the greater the buoyant force.
- Object's shape and orientation: These influence the submerged volume and, consequently, the upthrust.
2. Real-World Applications of Upthrust
Understanding buoyant forces is crucial in various fields, including:
- Shipbuilding: Designing ships to ensure buoyancy and stability.
- Submarine engineering: Controlling buoyancy for submerged and surfaced states.
- Hydrometry: Measuring fluid densities using objects' buoyant properties.
- Fluid mechanics: Analyzing forces on submerged structures and objects.
3. Limitations and Assumptions
- The calculation assumes the object is fully immersed and the fluid is incompressible and at rest.
- Effects of fluid viscosity are neglected; only buoyant force and weight are considered.
- The density of the solid object is not specified; the focus is on buoyant force derived from weight differences.
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Summary of Key Calculations
| Parameter | Value | Explanation |
|------------|--------|--------------|
| Actual weight in air (\(W_{air}\)) | 64 N | Given |
| Apparent weight in liquid (\(W_{liquid}\)) | 48 N | Given |
| Upthrust | 16 N | Calculated as \(64\, \text{N} - 48\, \text{N}\) |
| Density of liquid (\(\rho_{liquid}\)) | 800 kg/m³ | Converted from 0.8 g/cm³ |
| Volume of displaced fluid (\(V_{displaced}\)) | 0.00204 m³ | Calculated via upthrust relation |
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Conclusion
The calculation of the upthrust acting on a solid object immersed in a liquid provides essential insights into fluid mechanics principles. In this scenario, the object experiences an upthrust of 16 N when fully submerged in a liquid with a density of 0.8 g/cm³. This force corresponds to the weight of the displaced fluid, as per Archimedes’ principle. Understanding such principles has practical applications in engineering, design, and scientific research, emphasizing the importance of buoyancy in real-world contexts.
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Additional Resources and Further Reading
- Archimedes’ Principle and Its Applications
- Fluid Mechanics Fundamentals
- Density and Specific Gravity in Fluids
- Designing Buoyant Structures and Marine Vehicles
- Experiments Demonstrating Buoyancy and Upthrust
If you have further questions or need clarification on related topics, consider consulting physics textbooks or reputable online resources focused on fluid mechanics and buoyancy principles.
